Jump Discontinuity/Examples

Examples of Jump Discontinuities

Example 1

Let $f: \R \to \R$ be the real function defined as:

$\forall x \in \R: \map f x = \begin {cases} 1 & : x < 2 \\ 2 & : x \ge 2 \end {cases}$

Then $f$ has a jump discontinuity at $x = 2$.

In this case, $\map f 2$ is defined, and equals the limit from the right.

Example 2

Let $f: \R \to \R$ be the real function defined as:

$\forall x \in \R: \map f x = \begin {cases} 0 & : x > 1 \\ 1 & : x < 1 \\ \dfrac 1 2 & : x = 1 \end {cases}$

Then $f$ has a jump discontinuity at $x = 1$.

In this case, $\map f 1$ is defined, but equals neither limit.

Example 3

Let $f: \R \to \R$ be the real function defined as:

$\forall x \in \R: \map f x = \begin {cases} -1 & : x < 0 \\ 1 & : x > 0 \\ \text {undefined} & : x = 0 \end {cases}$

Then $f$ has a jump discontinuity at $x = 1$.

In this case, $\map f 0$ is not defined.